Question

A boat goes 12 km downstream and returns moving upstream to the same spot after 4 1/2

hours. The speed of the current is 2 km/h. Find the speed of the boat in still water.
​

Answer 1

$\large\underline{\sf{Solution-}}$

Let speed of the boat in still water be x km/hr.

Given that, speed of current = 2 km/hr.

Thus,

• Speed of upstream = x - 2 km/hr

• Speed of downstream = x + 2 km/hr

Case :- 1

Distance covered in upstream = 12 km

Speed of upstream = x - 2 km/hr

So,

Time taken in upstream to covered 12 km is

$\rm :\longmapsto\:t_1 = \dfrac{12}{x - 2} \: hr - - - (1)$

Case :- 2

Distance covered in downstream = 12 km

Speed of downstream = x + 2 km/hr

So,

Time taken in downstream to covered 12 km is

$\rm :\longmapsto\:t_2 = \dfrac{12}{x + 2} \: hr - - - (2)$

According to statement,

$\rm :\longmapsto\:t_1 \: + t_2 \: = \: \dfrac{9}{2}$

$\rm :\longmapsto\:\dfrac{12}{x - 2} + \dfrac{12}{x + 2} = \dfrac{9}{2}$

$\rm :\longmapsto\:\dfrac{12(x + 2) + 12(x - 2)}{(x - 2)(x + 2)} = \dfrac{9}{2}$

$\rm :\longmapsto\:\dfrac{12x + 24 + 12x - 24}{(x - 2)(x + 2)} = \dfrac{9}{2}$

$\rm :\longmapsto\:\dfrac{24x}{ {x}^{2} - 4 } = \dfrac{9}{2}$

$\rm :\longmapsto\: {9x}^{2} - 36 = 48x$

$\rm :\longmapsto\: {9x}^{2} - 36 - 48x = 0$

$\rm :\longmapsto\: {9x}^{2} - 48x - 36 = 0$

$\rm :\longmapsto\:3( {3x}^{2} - 16x - 12) = 0$

$\rm :\longmapsto\:{3x}^{2} - 16x - 12 = 0$

$\rm :\longmapsto\:{3x}^{2} - 18x + 2x - 12 = 0$

$\rm :\longmapsto\:3x(x - 6) + 2(x - 6) = 0$

$\rm :\longmapsto\:(x - 6)(3x + 2) = 0$

$\rm :\implies\:x = 6 \: \: \: \: or \: \: \: \: x = - \dfrac{2}{3} \: \: \red{(rejected)}$

Hence,

$\underbrace\blue{\boxed{ \bf \: Speed \: of \: boat \: in \: still \: water \: is \: 6 \: km \: per \: hr}}$

Answer 2

Let speed of the boat in still water be x km/hr.

• Given that, speed of current = 2 km/hr.

Thus,

• Thus,Speed of upstream = x - 2 km/hr

• Speed of downstream = x + 2 km/hr

CASE 1 : -

Distance covered in upstream = 12 km

Speed of upstream = x - 2 km/hr

So,

Time taken in upstream to covered 12 km is

$\\ \\ \\ \\ = \frac{12}{x - 2} \: \: hr \: - - - (i)$

CASE :- 2

Distance covered in downstream = 12 km

Speed of downstream = x + 2 km/hr

So,

Time taken in downstream to covered 12 km is

$\\ \\ \\ = \frac{12}{x + 2} \: \: \: hr - - - (ii)$

NOW,

ACCORDING TO THE QUESTION

$\\ \\ \\ \\ t1 \: \: + \: \: t2 = \frac{9}{2} \\ \\ \\ \\ \frac{12}{x - 2} + \frac{12}{x + 2} = \frac{9}{2} \\ \\ \\ \\ \frac{12(x + 2) + 12(x - 2)}{(x - 2)(x + 2)} = \frac{9}{2} \\ \\ \\ \\ \frac{12x + 24 + 12x - 24}{ {x}^{2} - 4} = \frac{9}{2} \\ \\ \\ \\ 24x(2) = 9( {x}^{2} - 4) \\ \\ \\ \\ 48x = 9 {x}^{2} - 36 \\ \\ \\ \\ 9 {x}^{2} - 48x - 36 = 0 \\ \\ \\ \\ 3(3 {x}^{2} - 16x - 12) = 0 \\ \\ \\ \\ 3 {x}^{2} - 18x + 2x - 12 = 0 \\ \\ \\ \\ 3 x(x - 6) \: \: + \: \: 2(x - 6) = 0 \\ \\ \\ \\ (3x + 2) \: \: (x - 6) = 0$

Hence ,

Either -----

$\\ \\ \\ \\ \\ x = \frac{ - 2}{3} \: \: \: \: \: (rejected)\: \: \: \\ \\ \\ \\ or \: \\ \\ \\ \: x = 6 \: \:$

HENCE, SPEED OF BOAT IN STILL WATER IS 6 KM/HR